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AI: Representation and Problem Solving

First-Order Logic

Instructors: Pat Virtue & Stephanie Rosenthal

Slide credits: CMU AI, http://aima.eecs.berkeley.edu

Outline

Need for first-order logic

Syntax and semantics

Planning with FOL

Inference with FOL

Pros and Cons of Propositional Logic

Propositional

logic

is

declarative

:

pieces

of

syntax

correspond

to

facts

Propositional

logic

allows

partial/disjunctive/negated

information

(unlike

most

data structures

and

databases)

Propositional

logic

is

compositional

:

meaning

of

B

1

,

1

∧

P

1

,

2

is

derived

from

meaning

of

B

1

,

1

and

of

P

1

,

2

Meaning

in

propositional

logic

is

context-independent

(unlike

natural

language,

where

meaning

depends

on

context)

Propositional

logic

has

very

limited

expressive

power

(unlike

natural

language)

E.g.,

cannot

say

“pits

cause

breezes

in

adjacent

squares”

except

by

writing

one

sentence

for

each

square

Pros and Cons of Propositional Logic

Conciseness

We don’t need to write out the successor-state axioms for each state individually, we can use variables and qualifiers

Rules of chess:

100,000 pages in propositional logic

1 page in first-order logic

Rules of

pacman

:



x,y,t

At(

x,y,t

)

 [

At(x,y,t-1)

 

u,v

Reachable(

x,y,u,v,Action

(t-1))] v

[

u,v

At(u,v,t-1)

 Reachable(

x,y,u,v,Action

(t-1))]

First-Order Logic (First-Order Predicate Calculus)

Whereas

propositional

logic

assumes

world

contains

facts

,

first-order

logic

(like

natural

language)

assumes

the

world

contains

Objects

:

people,

houses,

numbers,

theories,

Ronald

McDonald,

colors,

baseball

games, wars,

centuries,

...

Relations (return true/false)

:

red,

round,

bogus,

prime,

multistoried

..

.

,

brother

of,

bigger

than,

inside,

part

of,

has

color,

occurred

after,

owns,

…

Functions (return an object)

:

father

of,

best

friend,

third

inning

of,

one more

than,

end

of, …

Logics in General

LanguageWhat exists in the worldWhat an agent believes about factsPropositional logicFactstrue / false / unknownFirst-order logicfacts, objects, relationstrue / false / unknownProbability theoryfactsdegree of beliefFuzzy logicfacts + degree of truthknown interval value

Syntax of FOL

Basic Elements

Constants Predicates Functions Variables Connectives Equality Quantifiers

KingJ ohn, 2, CMU, . . .Brother, >, . . .Sqrt, LeftLegOf, . . . x, y, a, b, . . .∧ ∨ ¬ ⇒ ⇔=∀ ∃

Syntax of FOL

Atomic sentence = predicate(term1, . . . , termn) or term1 = term2 Term = function(term1, . . . , termn) or constant or variableExamplesBrother(KingJ ohn, RichardT heLionheart)> (Length(LeftLegOf (Richard)), Length(LeftLegOf (KingJ ohn)))

Syntax of FOL

Complex sentences are made from atomic sentences using connectives¬S, S1 ∧ S2, S1 ∨ S2, S1 ⇒S2, S1 ⇔ S2Examples Sibling(KingJ ohn, Richard) ⇒ Sibling(Richard, KingJ ohn)>(1, 2) ∨ ≤(1, 2)>(1, 2) ∧ ¬>(1, 2)

Models for FOL

Example

R

J

$

left

leg

left leg

brother

brother

person

on headperson king

crown

Models for FOL

Brother(Richard, John)Consider the interpretation in which:Richard → Richard the Lionheart John → the evil King JohnBrother → the brotherhood relationWhat does the Brother relationship mean?

R

J

$

left

leg

left leg

brother

brother

person

on head person king

crown

Model for FOL

Lots of models!

R

J

$

left

leg

left

leg

brother

brother

person

on head person king

crown

Model for FOL

Lots of models!Entailment in propositional logic can be computed by enumerating modelsWe can enumerate the FOL models for a given KB vocabulary: For each number of domain elements n from 1 to ∞For each k-ary predicate Pk in the vocabulary For each possible k-ary relation on n objectsFor each constant symbol C in the vocabularyFor each choice of referent for C from n objects . . .Computing entailment by enumerating FOL models is not easy!

Truth in First-Order Logic

Sentences

are

true

with

respect

to

a

model

and an

interpretation

Model

contains

≥

1

objects

(

domain

elements

)

and

relations

among

them

Interpretation

specifies

referents

for

constant

symbols

→

objects

predicate

symbols

→

relations

function

symbols

→

functional

relations

An

atomic

sentence

predicate

(

term

1

,

.

.

.

,

term

n

)

is

true

:

iff

the

objects

referred

to

by

term

1

,

. . . ,

term

n

are

in

the

relation

referred

to

by

predicate

Models for FOL

Consider the interpretation in which:Richard → Richard the Lionheart John → the evil King JohnBrother → the brotherhood relationUnder this interpretation, Brother(Richard, John) is true just in the case Richard the Lionheart and the evil King John are in the brotherhood relation in the model

R

J

$

left

leg

left leg

brother

brother

person

on head person king

crown

Universal Quantification

∀ (variables) (sentence)Everyone at the banquet is hungry:∀ x At(x, Banquet) ⇒ Hungry(x)∀ x P is true in a model m iff P is true with x beingeach possible object in the modelRoughly speaking, equivalent to the conjunction of instantiations of P(At(KingJ ohn, Banquet) ⇒ Hungry(KingJ ohn))∧ (At(Richard, Banquet) ⇒ Hungry(Richard))∧ (At(Banquet, Banquet) ⇒ Hungry(Banquet))∧ . . .

Universal Quantification

Common mistakeTypically, ⇒ is the main connective with ∀Common mistake: using ∧ as the main connective with ∀:∀ x At(x, Banquet) ∧ Hungry(x)means “Everyone is at the banquet and everyone is hungry”

Existential Quantification

∃ (variables) (sentence)Someone at the tournament is hungry:∃ x At(x, Tournament) ∧ Hungry(x)∃ x P is true in a model m iff P is true with x beingsome possible object in the modelRoughly speaking, equivalent to the disjunction of instantiations of P(At(KingJ ohn, Tournament) ∧ Hungry(KingJ ohn))∨ (At(Richard, Tournament) ∧ Hungry(Richard))∨ (At(Tournament, Tournament) ∧ Hungry(Tournament))∨ . . .

Existential Quantification

Common mistakeTypically, ∧ is the main connective with ∃Common mistake: using ⇒ as the main connective with ∃:∃ x At(x, Tournament) ⇒ Hungry(x)is true if there is anyone who is not at the tournament!

Properties of Quantifiers

∀ x∀ yis the same as ∀ y∀ x∃ x∃ yis the same as ∃ y∃ x∃ x∀ yis not the same as∀ y∃ x

∃ x ∀ y Loves(x, y)“There is a person who loves everyone in the world”∀ y ∃ x Loves(x, y)“Everyone in the world is loved by at least one person”Quantifier duality: each can be expressed using the other

∀ x Likes(x, IceCream)∃ x Likes(x, Broccoli)

¬∃ x ¬Likes(x, IceCream) ¬∀ x ¬Likes(x, Broccoli)

Fun with Sentences

Brothers

are

siblings

∀

x,

y

Brother

(

x,

y

)

⇒

Sibling

(

x,

y

)

.

“Sibling”

is

symmetric

∀

x,

y

Sibling

(

x,

y

)

⇔

Sibling

(

y,

x

)

.

A

first

cousin

is

a

child

of

a

parent’s

sibling

∀

x,

y

FirstCousin

(

x,

y

)

⇔

∃

p,

ps

Parent

(

p,

x

)

∧

Sibling

(

ps

,

p

)

∧

Parent

(

ps

,

y

)

Equality

term1 = term2 is true under a given interpretationif and only if term1 and term2 refer to the same objectE.g., 1 = 2 and ∀ x × (Sqrt(x), Sqrt(x)) = x are satisfiable 2 = 2 is validE.g., definition of (full) Sibling in terms of Parent:∀ x, y Sibling(x, y) ⇔ [¬(x = y) ∧ ∃ m, f ¬(m = f ) ∧ Parent(m, x) ∧ Parent(f, x) ∧ Parent(m, y) ∧ Parent(f, y)]

Piazza Poll 1

Given the following two FOL sentences:Which of these is true? BothNeither

Piazza Poll 1

Given the following two FOL sentences:Which of these is true? BothNeither

Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KBand perceives a smell and a breeze (but no glitter) at t = 5:T ell(KB, P ercept([Smell, Breeze, N one], 5))Ask(KB, ∃ a Action(a, 5))i.e., does KB entail any particular actions at t = 5? Answer: Y esGiven a sentence S and a substitution σ,Sσ denotes the result of plugging σ into S; e.g.,S = Smarter(x, y)σ = {x/EVE, y/WALL-E}Sσ = Smarter(EVE, WALL-E)Ask(KB, S) returns some/all σ such that KB |= Sσ

, {a/Shoot} ← substitution (binding list)

Notation Alert!

Notation Alert!

Inference in First-Order Logic

A) Reducing first-order inference to propositional inferenceRemoving Removing UnificationB) Lifting propositional inference to first-order inferenceGeneralized Modus PonensFOL forward chaining

Universal Instantiation

Every instantiation of a universally quantified sentence is entailed by it:∀ v αSubst({v/g}, α)for any variable v and ground term gE.g., ∀ x King(x) ∧ Greedy(x) ⇒ Evil(x) yieldsKing(John) ∧ Greedy(John) ⇒ Evil(John) King(Richard) ∧ Greedy(Richard) ⇒ Evil(Richard)King(Father(John)) ∧ Greedy(Father(John))⇒ Evil(Father(John))

Existential Instantiation

For any sentence α, variable v, and constant symbol kthat does not appear elsewhere in the knowledge base: ∃ v αSubst({v/k}, α)E.g., ∃ x Crown(x) ∧ OnHead(x, J ohn) yieldsCrown(C1) ∧ OnHead(C1, J ohn)provided C1 is a new constant symbol, called a Skolem constant

Reduction to Propositional Inference

Suppose

the

KB

contains

just

the

following:

∀

x

K

in

g

(

x

)

∧

G

r

eed

y

(

x

)

⇒

E

v

i

l

(

x

)

King

(

J

ohn

)

Greedy

(

J

ohn

)

Brother

(

Richard,

J

ohn

)

Instantiating

the

universal

sentence

in

all

possible

ways,

we

have

King

(

J

ohn

)

∧

Greedy

(

J

ohn

)

⇒

Evil

(

J

ohn

)

K

in

g

(

R

icha

r

d

)

∧

G

r

eed

y

(

R

icha

r

d

)

⇒

E

v

i

l

(

R

icha

r

d

)

King

(

J

ohn

)

Greedy

(

J

ohn

)

Brother

(

Richard,

J

ohn

)

The

new

KB

is

propositionalized

:

proposition

symbols

are

King

(

J

ohn

)

,

Greedy

(

J

ohn

)

,

Evil

(

J

ohn

)

,

King

(

Richard

)

etc.

Reduction to Propositional Inference

Claim:

a

ground

sentence

∗

is

entailed

by

new

KB

iff

entailed

by

original

KB

Claim:

every

FOL

KB

can

be

propositionalized

so as

to

preserve

entailment

Idea:

propositionalize

KB

and

query,

apply resolution,

return

result

Problem:

with

function

symbols,

there

are

infinitely

many

ground

terms,

e.g.,

Father

(

Father

(

Father

(

J

ohn

)))

Theorem:

Herbrand

(1930).

If a

sentence

α

is

entailed

by

an

FOL

KB,

it

is

entailed

by

a

finite

subset

of

the

propositional

KB

Idea:

For

n

=

0

to

∞

do

create

a

propositional

KB

by

instantiating

with

depth-

n

terms

see

if

α

is

entailed

by

this

KB

Problem:

works

if

α

is

entailed,

loops

if

α

is not

entailed

Theorem:

Turing

(1936),

Church

(1936),

entailment

in

FOL

is

semidecidable

Problems with Propositionalization

Propositionalization seems to generate lots of irrelevant sentences. E.g., from∀ x King(x) ∧ Greedy(x) ⇒Evil(x)King(John)∀ y Greedy(y)Brother(Richard, John)it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant

Unification

We can get the inference immediately if we can find a substitution θsuch that King(x) and Greedy(x) match King(John) and Greedy(y)θ = {x/John, y/J ohn} worksUnify(α, β) = θ if αθ = βθ

p

qθKnows(J ohn, x)Knows(J ohn, Jane){x/Jane}Knows(J ohn, x)Knows(y, Sam ){x/Sam, y/J ohn}Knows(J ohn, x)Knows(y, M other(y)){y/J ohn, x/M other(J ohn)}Knows(J ohn, x)Knows(x, Sam )fail

Standardizing apart eliminates overlap of variables, e.g., Knows(z17, Sam)

Generalized Modus Ponens (GMP)

where

for all

Example

is is is is is is is GMP used with KB of definite clauses (exactly one positive literal) All variables assumed universally quantified

FOL Forward Chaining